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On Witt Vector Cohomology for Singular Varieties ON WITT VECTOR COHOMOLOGY FOR SINGULAR VARIETIES PIERRE BERTHELOT, SPENCER BLOCH, AND HEL´ ENE` ESNAULT Abstract. Over a perfect field k of characteristic p > 0, we con- struct a “Witt vector cohomology with compact supports” for sep- arated k-schemes of finite type, extending (after tensorisation with Q) the classical theory for proper k-schemes. We define a canonical morphism from rigid cohomology with compact supports to Witt vector cohomology with compact supports, and we prove that it provides an identification between the latter and the slope < 1 part of the former. Over a finite field, this allows one to compute con- gruences for the number of rational points in special examples. In particular, the congruence modulo the cardinality of the finite field of the number of rational points of a theta divisor on an abelian variety does not depend on the choice of the theta divisor. This answers positively a question by J.-P. Serre. Contents 1. Introduction 1 2. Witt vector cohomology with compact supports 6 3. A descent theorem 17 4. Witt vector cohomology and rigid cohomology 21 5. Proof of the main theorem 28 6. Applications and examples 33 References 41 1. Introduction Let k be a perfect field of characteristic p > 0, W = W (k), K = Frac(W ). If X is a proper and smooth variety defined over k, the theory of the de Rham-Witt complex and the degeneration of the slope spectral sequence provide a functorial isomorphism ([6, III, 3.5], [20, II, 3.5]) ∗ <1 ∼ ∗ (1.1) Hcrys(X/K) −−→ H (X, W OX )K , Date: October 12, 2005. 1 2 PIERRE BERTHELOT, SPENCER BLOCH, AND HEL´ ENE` ESNAULT ∗ <1 where Hcrys(X/K) is the maximal subspace on which Frobenius acts with slopes < 1, and the subscript K denotes tensorisation with K. If we only assume that X is proper, but maybe singular, we can use rigid cohomology to generalize crystalline cohomology while retaining all the standard properties of a topological cohomology theory. Thus, the left hand side of (1.1) remains well defined, and, when k is finite, the alternated product of the corresponding characteristic polynomials of Frobenius can be interpreted as the factor of the zeta function ζ(X, t) “of slopes < 1” (see 6.1 for a precise definition). On the other hand, the classical theory of the de Rham-Witt complex can no longer be directly applied to X, but the sheaf of Witt vectors W OX is still available. Thus the right hand side of (1.1) remains also well defined. As in the smooth case, this is a finitely generated K-vector space endowed with a Frobenius action with slopes in [0, 1[ (Proposition 2.10), which has the advantage of being directly related to the coherent cohomology of X. It is therefore of interest to know whether, when X is singular, (1.1) can be generalized as an isomorphism ∗ <1 ∼ ∗ (1.2) Hrig(X/K) −−→ H (X, W OX )K , ∗ <1 ∗ where Hrig(X/K) denotes the subspace of slope < 1 of Hrig(X/K). Our main result gives a positive answer to this question. More gener- ally, we show that a “Witt vector cohomology with compact supports” can be defined for separated k-schemes of finite type, giving cohomology ∗ spaces Hc (X, W OX,K ) which are finite dimensional K-vector spaces, endowed with a Frobenius action with slopes in [0, 1[. Then, for any such scheme X, the slope < 1 subspace of the rigid cohomology with compact supports of X has the following description: Theorem 1.1. Let k be a perfect field of characteristic p > 0, X a sep- arated k-scheme of finite type. There exists a functorial isomorphism ∗ <1 ∼ ∗ (1.3) Hrig,c(X/K) −−→ Hc (X, W OX,K ). This is a striking confirmation of Serre’s intuition [24] about the relation between topological and Witt vector cohomologies. On the other hand, this result bears some analogy with Hodge theory. Recall from [8] (or [12]) that if X is a proper scheme defined over C, then its Betti cohomology H∗(X, C) is a direct summand of its de Rham ∗ • ∗ cohomology H (X, ΩX ). Coherent cohomology H (X, OX ) does not exactly compute the corner piece of the Hodge filtration, as would be an exact analogy with the formulation of Theorem 1.1, but it gives an upper bound. Indeed, by [14], Proposition 1.2, and [15], Proof ∗ of Theorem 1.1, one has a functorial surjective map H (X, OX ) 0 ∗ grF H (X, C). WITT VECTOR COHOMOLOGY OF SINGULAR VARIETIES 3 The construction of these Witt vector cohomology spaces is given in section 2. If U is a separated k-scheme of finite type, U,→ X an open immersion in a proper k-scheme, and I ⊂ OX any coherent ideal such that V (I) = X \U, we define W I = Ker(W OX → W (OX /I)), and we ∗ show that the cohomology spaces H (X, W IK ) actually depend only on U (Theorem 2.4). This results from an extension to Witt vectors of Deligne’s results on the independence on the compactification for the construction of the f! functor for coherent sheaves [18]. When U varies, these spaces are contravariant functors with respect to proper maps, and covariant functors with respect to open immer- sions. In particular, they give rise to the usual long exact sequence relating the cohomologies of U, of an open subset V ⊂ U, and of the complement T of V in U. Thus, they can be viewed as providing a notion of Witt vector cohomology spaces with compact support for U. We define ∗ ∗ Hc (U, W OU,K ) := H (X, W IK ). Among other properties, we prove in section 3 that these cohomology spaces satisfy a particular case of cohomological descent (Theorem 3.2) which will be used in the proof of Theorem 1.1. The construction of the canonical homomorphism between rigid and Witt vector cohomologies is given in section 4 (Theorem 4.5). First, we recall how to compute rigid cohomology for a proper k-scheme X when there exists a closed immersion of X in a smooth formal W - scheme P, using the de Rham complex of P with coefficients in an appropriate sheaf of OP-algebras AX,P. When P can be endowed with a lifting of the absolute Frobenius endomorphism of its special fibre, a simple construction (based on an idea of Illusie [20]) provides a func- ˇ torial morphism from this de Rham complex to W OX,K . Using Cech resolutions, this morphism can still be defined in the general case as a morphism in the derived category of sheaves of K-vector spaces. Then we obtain (1.3) by taking the morphism induced on cohomology and restricting to the slope < 1 subspace. By means of simplicial resolu- tions based on de Jong’s theorem, we prove in section 5 that it is an isomorphism. The proof proceeds by reduction to the case of proper and smooth schemes, using Theorem 3.2 and the descent properties of rigid cohomology proved by Chiarellotto and Tsuzuki ([10], [27], [28]). We now list some applications of Theorem 1.1, which are developed in section 6. We first remark that it implies a vanishing theorem. If X is smooth projective, and Y ⊂ X is a divisor so that U = X \ Y is i affine, then Serre vanishing says that H (X, OX (−nY )) = 0 for n large 4 PIERRE BERTHELOT, SPENCER BLOCH, AND HEL´ ENE` ESNAULT and i < dim(X). If k has characteristic 0, then we can take n = 1 by i Kodaira vanishing. One then has H (X, OX (−Y )) = 0 for i < dim(X). It is tempting to view the following corollary as an analogue when k has characteristic p : Corollary 1.2. Let X be a proper scheme defined over a perfect field k of characteristic p > 0. Let I ⊂ OX be a coherent ideal defining a closed subscheme Y ⊂ X such that U = X \ Y is affine, smooth and equidimensional of dimension n. Then i i i r Hc(U, W OU,K ) = H (X, W I)K = H (X, W I )K = 0 for all i 6= n and all r ≥ 1. i Indeed, when U is smooth and affine, Hrig(U/K) can be identified with Monsky-Washnitzer cohomology [4], and therefore vanishes for i > n = dim(U). If moreover U is equidimensional, it follows by i Poincar´eduality [5] that Hrig,c(U/K) = 0 for i < n. So (1.3) implies i that H (X, W I)K = 0 for i < n. On the other hand, the closure U of i i U in X has dimension n, and H (X, W I)K = Hc(U, W OU,K ) does not change up to canonical isomorphism if we replace X by U. Therefore i H (X, W I)K = 0 for i > n. a When k is a finite field Fq, with q = p , Theorem 1.1 implies a statement about zeta functions. By ([16], Th´eor`emeII), the Lefschetz trace formula provides an expression of the ζ-function of X as the alternating product Y (−1)i+1 ζ(X, t) = Pi(X, t) , i i a where Pi(X, t) = det(1 − tφ|Hrig,c(X/K)), and φ = F denotes the Fq-linear Frobenius endomorphism of X. On the other hand, we define W i (1.4) Pi (X, t) = det(1 − tφ|Hc(X, W OX,K )), W Y W (−1)i+1 ζ (X, t) = Pi (X, t) .
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